Optimal. Leaf size=156 \[ \frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {427, 542, 537,
223, 212, 385, 211} \begin {gather*} \frac {\sqrt {d} \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3}+\frac {(b c-a d)^{5/2} \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 542
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx &=\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {\int \frac {\sqrt {c+d x^2} \left (c (4 b c-a d)+d (7 b c-4 a d) x^2\right )}{a+b x^2} \, dx}{4 b}\\ &=\frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {\int \frac {c \left (8 b^2 c^2-9 a b c d+4 a^2 d^2\right )+d \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 b^2}\\ &=\frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^3 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}+\frac {\left (d \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^3}\\ &=\frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (d \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 b^3}\\ &=\frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 160, normalized size = 1.03 \begin {gather*} \frac {b d x \sqrt {c+d x^2} \left (9 b c-4 a d+2 b d x^2\right )-\frac {8 (b c-a d)^{5/2} \tan ^{-1}\left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a}}-\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2073\) vs.
\(2(130)=260\).
time = 0.13, size = 2074, normalized size = 13.29
method | result | size |
risch | \(-\frac {d x \left (-2 b d \,x^{2}+4 a d -9 b c \right ) \sqrt {d \,x^{2}+c}}{8 b^{2}}+\frac {d^{\frac {5}{2}} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}}{b^{3}}-\frac {5 d^{\frac {3}{2}} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a c}{2 b^{2}}+\frac {15 \sqrt {d}\, \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) c^{2}}{8 b}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) a^{3} d^{3}}{2 b^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) a^{2} c \,d^{2}}{2 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) a \,c^{2} d}{2 b \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right ) c^{3}}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) a^{3} d^{3}}{2 b^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) a^{2} c \,d^{2}}{2 b^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}-\frac {3 \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) a \,c^{2} d}{2 b \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right ) c^{3}}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(1389\) |
default | \(\text {Expression too large to display}\) | \(2074\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 931, normalized size = 5.97 \begin {gather*} \left [\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3}}, -\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3}}, \frac {8 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + {\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3}}, -\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{a + b x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^{5/2}}{b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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